Contemporary deep learning systems store knowledge implicitly in distributed weights, making local updates, structural constraints, and causal interpretation difficult under continual change. This paper proposes a new learning substrate in which knowledge is represented as an explicit geometric state.

A torus-based semantic dynamics model that represents perception, structure, and prediction through the evolution of angular states on a multidimensional torus. Unlike models operating in unconstrained Euclidean latent spaces, PhaseBrain encodes all modalities—text, code, audio, images, and time series—as trajectories of phases and amplitudes, enabling a single probabilistic law of angular dynamics shared across channels.

PIF maps any data — tables, images, signals, actions — into a shared phase space. This allows systems to connect modalities, find recurring patterns, and return results back into discrete form.